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Theorem rexeqbii 2354
Description: Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
raleqbii.1 𝐴 = 𝐵
raleqbii.2 (𝜓𝜒)
Assertion
Ref Expression
rexeqbii (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒)

Proof of Theorem rexeqbii
StepHypRef Expression
1 raleqbii.1 . . . 4 𝐴 = 𝐵
21eleq2i 2120 . . 3 (𝑥𝐴𝑥𝐵)
3 raleqbii.2 . . 3 (𝜓𝜒)
42, 3anbi12i 441 . 2 ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))
54rexbii2 2352 1 (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒)
Colors of variables: wff set class
Syntax hints:  wb 102   = wceq 1259  wcel 1409  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-clel 2052  df-rex 2329
This theorem is referenced by: (None)
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